SLIDING MODE CONTROL BASED ON TDC AND UDE

Transcription

1 INERNAIONAL JOURNAL JOURNAL OF OF INFORAION AND AND SYSES SYSES SCIENCES E Volume Volume 3, Number 1, Number 1, Pages 1, Pages Institute for Scientific Computing and Information SLIDING ODE CONROL BASED ON DC AND UDE CHANDRASEKHAR AND LILLIE DEWAN Abstract his paper also proposes another robust control strategy based on uncertainty and disturbance estimator (UDE)It brings similar performance as the time-delay control (DC), does not use delay in the system, no oscillations in the control signal and there is no need to measure the derivatives of the state vector he robust stability of LI-SISO is analyzed and the simulation results indicate the effectiveness of sliding mode control based on UDE with a comparison made with sliding mode control based on DC Key Words, Partner selection, Decision making method, Agile enterprise, Empirical study 1 INRODUCION ost of the well-developed control theory, either in the frequency domain or in the time domain, deals with systems whose mathematical representations are completely known However, in many practical situations, the parameters of the system are either poorly known or operate in environments where unpredictable large system parameter variations and unexpected disturbances are possible In such situations, the usual fixed-gain controller will be inadequate to achieve satisfactory performance in the entire range over which the characteristics of the system may vary Several advanced control techniques have been developed for such systems One of the primary methods is Sliding mode control (SC) which can deals with both linear and nonlinear systems Based on Lyapunov s method, the control scheme is characterized by discontinuous function with high frequency chattering he plant parameter variations and disturbances are assumed to be bounded [1, 4, 6] Another control method, ime Delay Control (DC) [11, 13, 14], depends on the direct estimation of the function representing the effect of uncertainties and disturbances overcoming the defects of chattering, bounded uncertainties and disturbances as in sliding mode control he DC employs past observation of the system response and control input to directly modify the control actions rather than adjusting the controller gains or identifying system parameters thereby leading to a model independent controller his algorithm can deal with large unpredictable system parameter variations and disturbances Although DC has very good potential to improve system performance; but, it is still desirable to get rid of delays, if possible, so as to simplify system analysis he assumption used in this paper for DC is a time-domain assumption An alternative control strategy has been proposed in the frequency domain to obtain similar performance to DC he two inherent drawbacks of DC, namely, there is no need to Received by the editors June 21,

2 SLIDING ODE CONROL BASED ON DC AND UDE 37 measure the derivative of the states and there is no oscillation in the control signal, are ad-dressed using a UDE control strategy [3] Also, an additional benefit is that the delay is removed he system stability is more easily analyzed In addition to all these advantages, the system performance is very similar to that obtained using DC his paper focuses on the implementation of SC, SCDC and SCUDE under different conditions and comparing their performance In section 2 sliding mode control, sliding condition, chattering effect are discussed Section 3 discusses the ime Delay control law, Design of sliding mode control based on DC and the results are compared with the conventional sliding mode control Section 4 deals with UDE based control law, design of sliding mode control based on Uncertainty and Disturbance estimator he results confirm that UDE follows the model very closely in spite of significant uncertainties and disturbances and finally paper concludes in section 6 2 SLIDING ODE CONROL Sliding mode control (SC) is a special case of variable structure system SC is a robust control method which can handle both linear and nonlinear systems with parametric uncertainties and unexpected disturbances It is based on the concept of steering the states of a system to a specified stable manifold and using high speed switching to maintain the subsequent motion on this manifold he control input is chosen such that the trajectory near the sliding surface is directed towards it Once the system is trapped on the surface, closed loop dynamics are completely governed by the equation and parameters that define the surface hus, closed loop dynamics of the system will be independent of perturbations in the parameters of the system as the parameters defining the surface are chosen by the designer Due to the requirement for instantaneous control action SC has been developed in the continuous-time domain to a great extent [1, 2, 4, 6] Design of a sliding mode control law can be broken into three steps 1 Design an equilibrium surface called sliding surface σ = SX = 0, such that state trajectory of the system restricted to the sliding surface is characterized by the desired behavior 2 Choose the control law such that the trajectories near the surface points towards the surface 3 Determine the system dynamics on the surface he basic problems then are, the specification of the gains in the controller so that the motion trajectory reaches the surface in a finite time, the determination of the switching logic of the controller so that the motion of the system is constrained to the surface and equation defining the surface that dictates the dynamic behavior of the system on the surface [1, 5] 21 SLIDING CONDIION Consider a state space representation of a plant X = AX + Bu (1)

3 38 CHANDRASEKHAR AND L DEWAN he objective of SC is to force the plant states towards the sliding surface σ = SX (2) Lyapunov stability criteria Lyapunov s stability criteria states that for system equation if there exists a scalar function V (x), called Lyapunov function, which for some real number > 0 satisfies the following properties for all x in the region X < 1 V (x) = 0 if x = 0 2 V (x) > 0 if x 0 3 V (x) has continuous partial derivatives wrt all x hen the equilibrium point is 1 Asymptotically stable if V (x) < 0; 2 Asymptotically stable in the large if V (x) < 0; V (x) as X Consider now a Lyapunov function V (x) =½σ 2 (3) As per the Lyapunov stability criteria stated above, the system equation will have a stable equilibrium point if V ( X ) 0 (4) hus the condition for stability becomes V = σ σ (5) σ σ 0 (6) his is known as the sliding condition hus, by designing a control law that satisfies stability condition given by eq (6), an unstable system or marginally stable system can be stabilized and its behavior can be made invariant to the unknown system perturbations and disturbances In the sliding phase however, the system response is completely invariant to the system uncertainties and disturbances [2, 7] 22 CHAERING EFFEC Sliding mode control implies that control action is discontinuous as shown in Fig2, corresponding Simulink block diagram is shown in Fig1 and since the implementation of associated control switching cannot be perfect ie the switching frequency is limited in real system, this will cause oscillations of high frequency which is usually referred to as chattering hus, the practical implementation of classical sliding mode controllers results in the control chattering In practice a small, but non zero delay in the control switching will cause the trajectory to slightly overshoot the switching surface each time the control is switched Larger the discontinuity the more severe the control chatter will be Chattering involves high control activity and may excite unmodeled dynamics [1, 4]It is also responsible for wear and tear of actuators as in the case of hydraulic actuators Chattering is therefore, an undesirable phenomenon and should be eliminated his paper highlights another solution to control chattering by using concept of ime-delay control (DC)

4 SLIDING ODE CONROL BASED ON DC AND UDE 39 FIG: 1 SLIDING ODE CONROL FIG: 2 RESPONSE OF CONVENIONAL SC 3 SLIDING ODE CONROL BASED ON DC: ime Delay Control employs direct estimation of the effect of the plant dynamics through the use of time delay It does not depend on the estimation of specific parameters of the system he gathered information is used to cancel the unknown dynamics and the unexpected disturbances simultaneously and then the controller inserts

5 40 CHANDRASEKHAR AND L DEWAN the desired dynamics into the plant he DC employs past observations of the system response and control inputs to directly modify the control actions Using DC with SC, the draw backs of SC should be overcome and the assumptions made on the uncertainties are also relaxed [13, 14] 31 DESIGN OF SCDC he system to be considered is formulated as X = ( A+ A) X ( + ( B + B) u( + Fd( X, (7) Where A, B are uncertainties in A, B respectively d(x, is the unknown external disturbance and the variable t represents the time A, B are constant matrices with appropriate dimensions A = BD, B=BE, F = Bf (8) are the matching conditions Using Eq(8) in Eq(7) X ( = ( A+ BD) X ( + ( B + BE) u( + Bfd( X, hen X ( = AX ( + Bu( + Be( X, (9) Where e(x, = AX( + Bu( + fd(x, (10) Sliding surface : σ= SX( (11) he Lyapunov function is defined as: he sliding condition is 2 V ( X ) = 1/ 2σ (12) V = σ σ 0 (13) From Eq (11) and Eq(9) σ = ( S X) = SA X( + SBu( + SBe( X, (14) Control law is defined as u( = u eq + u un (15) from eq(15)and eq(14), σ = SAX ( + SB( ueq + uun ) + SBe( X, when σ =0 1 ( SB) ( SAX ( ) u eq = (16) then σ = SBe ( X, + SBu un (17) Here the value of e(x, is not known for calculating u un For estimating the value of e(x, ime-delay control technique is used he basic assumption is that e(x, does not change significantly over a small interval L (ime-delay) So e(x, e(x, t L) then Eq(14), is shifted one step backward, e( X, t L) = ( SB 1 )( σ ( t L) SAX( t L) SBu( t L)) (18)

6 SLIDING ODE CONROL BASED ON DC AND UDE 41 From eq(17)and eq(18) u un is evaluated as follows u un = ( SB 1 4 SLIDING ODE CONROL BASED ON UDE )( σ ( t L) SAX( t L) SBut ( L)) (19) A new design of sliding mode control based on the uncertainty and disturbance estimator (UDE) is used without the knowledge of the bounds of uncertainties and disturbances and it is continuous, thus two main difficulties in the design of sliding mode control are overcome [3, 12] Further the method of uncertainty and disturbance estimator is extended to plants having significant uncertainty in the control input matrix and subjected to disturbances that depend on states nonlinearly In the conventional sliding mode control systems, insensitivity to uncertainties and disturbances is guaranteed by employing a discontinuous control based on the bounds of uncertainties and disturbances[1] In many situations these bounds are hard to find, resulting in an overestimation and consequently a large control he discontinuous control is objectionable because it can cause excessive wear and tear of actuators and may excite unmodeled dynamics A common concern in all these methods is the need for the derivative of the state or the sliding surface variable Recently Zhong and Rees [3] have proposed a very promising method called the uncertainty and disturbance estimator (UDE) for control of linear time invariant systems his method, based on a concept similar to the DC, does not require the derivative of the system state and does not use time delayed signals he assumption used in DC is a time domain assumption An assumption in the frequency domain to propose an alternative control strategy (UDE) to obtain similar performance to DC has been used In this section, the results are extended in two ways First, the method of UDE is applied to sliding mode control so as to control uncertain systems without having to use a discontinuous control and without needing the bounds on the uncertainties and disturbances Secondly, the plant considered has uncertainty in the control input matrix and the disturbance term contains state dependent nonlinearities hen SC with UDE method has been compared with that of SC with DC 41 DESIGN OF SCUDE: he system described in SCDC eq(7) rewritten as X = ( A+ A) X + ( B + B) u + Fd( X, (20) Where X is the state vector, u is the control input, A and B are known constant matrices, A, B are uncertainties and d(x, is an unknown disturbance Assumption 1: he uncertainties A, B and the disturbance d(x, satisfy the matching

7 42 CHANDRASEKHAR AND L DEWAN conditions given by: A = BD, B = BE, Fd(x, = Bv(x, (21) Where D and E are unknown matrices of appropriate dimensions and v(x, is an unknown function he system (20) can now be written as X ( = AX + Bu + Be( X, (22) Where e(x, = Dx + Eu + v(x, Although e(x, contains uncertainty and disturbances, it will be referred to as the lumped uncertainty for the sake of convenience Next let X + A X + B u be a stable model satisfying certain structural conditions stated by the following assumption: Assumption 2: he choice of the model is such that A A = BL, B = B Where L and are suitable known matrices he objective is to design a control u so as to force the system Eq(20) to follow the model Eq(23) in spite of the uncertainties and disturbances represented by e(x, (23) Design of control In this section a model following control is designed in the framework of sliding mode control he method is based on a sliding surface suggested in [4] and is different from the sliding mode control based on model following reported in literature Define a sliding surface σ = b t x + z (24) where Z = B A X B B u ; Z( 0) = B X(0) (25) he equation Eq(25) for the auxiliary variable z given here is different from that given in [12]By virtue of the choice of the initial condition on z, σ = 0 at t = 0 If a control u can be designed ensuring sliding then σ = 0 implies X = A X + B u ` (26) which fulfills the objective of model following Differentiating Eq(24) and using Eq(22) with Eq(25) σ = B AX + B Bu + B Be( X, B A X B B u (27)

8 SLIDING ODE CONROL BASED ON DC AND UDE 43 B BLX B Bu+ B Bu+ B BeX (, = (28) Let the required control be expressed as u = u eq + u n (29) Selecting u eq = Lx +u m (30) from Eq(27), Eq(29) and Eq(30), σ = B Next the component u n will be designed Bu n + B Be( X, (31) 42 COPENSAION OF UNCERAINIES AND DISURBANCES he lumped uncertainty e(x, can be compensated by estimating it as in [2] except that instead of using the state equation Eq(22),equation Eq(30) is used for σ Rewriting Eq(31) as 1 e( X, = σ u n (32) B B it can be seen that the lumped uncertainty e(x, can be computed from the right hand side of Eq(32) his however cannot be done directly Let the estimate of the lumped uncertainty, denoted by, ê (x,, be defined as 1 e ˆ( X, = sg f ( s) σ G f ( s) un (33) B B where G f (s) is a strictly proper low-pass filter with unity steady-state gain and broad enough bandwidth and sg f is physically implement able and there is no need for measuring the derivative of states With such a filter ê (x, e(x, (34) enabling the design of u n u n = (x, (35) ê Solving for u n gives u n 1 = sg ( s)σ (36) f B B(1 G ( s)) f Clearly, since G f (s) is strictly proper the control signal u n in Eq(36) is implement able 43 EXISENCE OF SLIDING ODE he existence of the sliding mode can be proved easily Define the error in estimation as eˆ ( X, e( X, eˆ( X, = (37)

9 44 CHANDRASEKHAR AND L DEWAN Using Eq(35) in Eq(31), we get σ = B Beˆ( X, (38) which, in view of Eq(34), leads to σσ = 0 (39) Since σ = 0 at t = 0 by virtue of the choice of σ in Eq(24) and Eq(25), satisfaction of Eq(39) ensures σ = 0for all t 0 his makes the uncertain plant follow the stable model chosen by the designer for all t 0 44 Accuracy of estimation he above result is based on the premise that Eq(34) holds Consider a practical low-pass filter 1 G f ( s) = (40) s + 1 where is a small positive constant With the above G f (s) and in view of Eq(32) and Eq(33) 1 eˆ ( X, = (1 G f ( s)) ( σ + kσ ) u = e (x, G n B B f (s) (41) herefore Eq(34) will hold, if the term e (x, is sufficiently small Interestingly, this is similar to the usual assumption e(x, e(x, t L), where L is a small interval of time, found in the DC approach It is worth noting that for Gf (s) in Eq(40) the control u n works out to 1 un = σ (42) B B and has a simple time domain interpretation From Eq(41) and Eq(42), it is clear that smaller implies a smaller estimation error but a larger magnitude of control if sigma is not small he choice of σ as given in Eq(24) and Eq(26), enables the designer to strike a favorable compromise in this respect 5 CASE SUDY COPARISON OF SC, SCDC Simulation was done for the following second order system with a single input of the form in eq(7)which is rewritten as X = ( A+ A) X ( + ( B + B) u( + Fd( X, to demonstrate the effectiveness of SCDC over SC using ALAB 704System matrices are A= 0 1,B= 0,C= 0,D= [ 0 0 ] with uncertainties in A and B as A= 0 0 ; B= 0 ; and the initial condition as X 0 = 1,choosing the sliding surface as S= [ 4 1] so that it 0

10 SLIDING ODE CONROL BASED ON DC AND UDE 45 satisfies the Lyapunov condition for asymptotic stability he corresponding control signal and the states of the system are obtained as shown in Fig2 and the corresponding Simulink block diagram of conventional sliding model control is shown in Fig1 he results indicate that there is chattering in the control signal and as the uncertainties and disturbances are increased, it worsens the case severely For the same system as described above the DC has been used along with sliding mode control by choosing delay time (L) as 100ms, the error feedback gain matrix as 4 and the disturbance is of amplitude 01 with a frequency of 1as shown in Simulink block diagram Fig3 FIG: 3 SIULINK BLOCK DIAGRA OF SC BASED ON DC From the results of Fig4, indicates chattering in the control signal of conventional sliding mode control has been nullified and the time for the convergence of states to the desired trajectory is small in SCDC and the assumption on unexpected disturbances and uncertainties in SC has been relaxed in SCDC

11 46 CHANDRASEKHAR AND L DEWAN FIG: 4 RESPONSE FOR SC BASED ON DC For SCDC, an external noise has been added to both the states whose amplitude is unity and frequency of 60rad/sec and results are obtained as shown in Fig5and Fig6 FIG: 5 RESPONSE FOR SC BASED ON DC WIH NOISE

12 SLIDING ODE CONROL BASED ON DC AND UDE 47 FIG: 6 RESPONSE FOR SC BASED ON DC WIH NOISE he simulation results indicate that there are oscillations of constant magnitude in the control signal he states of the system converge to the sliding surface after t=12 with one of the states having oscillations of constant amplitude he same noise has been applied to the same system by filtering with least square filter of time constants 1 =1/150, 2=1/178 and the results indicates damped oscillations in the control signal in one of the states with a convergence time of t=09, but the computation time of SCDC is very large he comparison is summarized in able 1 able-1 ethod of control Control signal Convergence of the states after a time(t=) Sliding mode chattering 16 control SCDC Smooth control 14 SCDC with Results in oscillations of 12with one of the states having noise constant amplitude oscillations of constant amplitude SCC with filtering Noise Damped oscillations 09 with one of the states having damped oscillations

13 48 CHANDRASEKHAR AND L DEWAN COPARISON OF SC, SCDC, SCUDE: Consider a second order system with a single input of the form in eq(7) rewritten as X = ( A+ A) X ( + ( B + B) u( + Fv( X, to show the comparison of the three methods Let the system matrices be A= 0 1, B= 0, C= 1 0, D =[ 1 0 0]; X0 =, A= 0 0, B= 0 where A, B are the uncertainties in system matrices A and B, D is the external 0 disturbance 1 with initial condition as X0, and the corresponding model matrices are A m = 0 1 ; B m= ; C m= ; X0= 0 ; =0001, L= Where X 0 is the initial model state, is the time constant of the low pass filter used in SCUDE, L is the delay used in SCDC Choosing the sliding surface coefficient vector as S=[ 4 1] and feedback gain matrix K=5 he disturbance term for this case 2 is given by v(x, = 2(sin(10 x1 +cos(10 x2 +1) he Simulink block diagrams for SC, SCDC and SCUDE are shown in Fig7, Fig9 and Fig12 respectively From the simulation results of Fig8(SC), Fig10(SCDC) and Fig13(SCUDE) indicates that there is no chattering,no oscillations in the control signal and the time for the convergence of the states to desired trajectory is small in SCUDE as compared to SCDC and SC No need to introduce delay in the system as compared to SCDC and the performance is similar to SCDCBy using SC with DC for the same system as explained above, it appears that the method of UDE copes with fast varying disturbances and large control input matrix uncertainties better than the DC [12] as shown in Fig 13 FIG: 7 SIULINK BLOCK DIAGRA OF SC WIH ODEL REFERENCE

14 SLIDING ODE CONROL BASED ON DC AND UDE 49 Fig: 8 RESPONSE FOR SC WIH ODEL REFERENCE Fig: 9 SIULINK BLOCK DIAGRA FOR SC WIH DC Fig: 10 RESPONSE FOR SC BASED ON DC

15 50 CHANDRASEKHAR AND L DEWAN Fig: 12 SIULINK BLOCK DIAGRA OF SC BASED ON UDE Fig: 13 RESPONSE FOR SC BASED ON UDE Finally a variable noise of 60rad/sec has been introduced to SCDC and SCUDE and the results are observed as in Fig11& Fig14

16 SLIDING ODE CONROL BASED ON DC AND UDE 51 FIG: 11 RESPONSE FOR SC BASED ON DC WIH NOISE Fig: 14 RESPONSE FOR SC BASED ON UDE WIH NOISE It is clear that in SCUDE there is oscillations of constant amplitude in the control signal and one of the states has damped oscillations with a converging time t=2,but SCDC has large oscillations of constant amplitude in the control signal and one of the states has oscillations of constant amplitude Computation time of SCDC is large compared to SCUDE Finally the results are summarized in able-2

17 52 CHANDRASEKHAR AND L DEWAN able-2 ethod of control Control signal Convergence of the states after a time(t=) Sliding mode control Chattering 32 but not good model following SCDC Smooth control but small oscillations, computation time is large 24 SCUDE Smooth control 1 SCUDE Noise SCDC Noise with with Oscillations of constant amplitude Large Oscillations of constant amplitude 2 with one of the states having oscillations of constant amplitude 24 with one of the states having damped oscillations CONCLUSION his paper discussed the control of systems with unknown dynamics and unexpected disturbances ethods of SCDC and SCUDE were proposed for the control of second order system SC results in a discontinuous law and also an estimated value of control depending on the amount of uncertainty and disturbance It has been shown that, using SCUDE without measuring the derivative of the states has eliminated the inherent drawbacks of the SCDC; oscillation in the control signal has been eliminated It is easier to analyze the stability of SCUDE he same work can be extended for relative degree greater than one and developing SC based on DC+UDE using only output of the system without depending on the states of the system REFERENCES 1 Ackermann, J and Utkin, V I, Sliding mode control design based on Ackermanns formula, IEEE rans Automat Contr, Vol 43, 1998, pp Bo Feng, C and Wu, Y-Q, A Design Scheme of Variable Structure Adaptive control for Uncertain Dynamic Systems, IEEE rans on Automatic control, Vol 32, 1996, pp CQ Zhong and Rees D, Control of uncertain LI systems based on an uncertainty and disturbance estimator, Journal of Dynamic Systems and easurements Cont, Vol 126, 2004, pp Ching-Chang Wong and Shih-Yu Chang, Parameter Selection in the Sliding ode Control Design Using Genetic Algorithms amkang Journal of Science and Engineering, vol1, No 2, 1998,pp Corradini, and OrlandoG, Variable Structure control of Discretized Continuous ime systems, IEEE rans on Automatic Control, Vol 43, September 1998, pp Elmali, H and Olgac, N, Sliding mode control with perturbation estimation (SCPE): a new approach, Int Journal of Control, Vol 56, 1992, pp

18 SLIDING ODE CONROL BASED ON DC AND UDE 53 7 H orioka K W and Sabanovic A, Sliding mode control based on the time delay estimation, IEEE Workshop on Variable Structure Systems, 1996, pp HiroshiKW and AsifS, Sliding ode control based on the ime Delay estimation, IEEE workshop on Variable Structure Systems, 1996, pp IoannouPA and PV Kokotovic, Instability analysis and improvement of robustness of adaptive control, Automatic control, Vol20, 1984, pp KreisselmeierG and BDO Anderson, Robust model reference adaptive control, IEEE rans on Automatic contro,vol-31, 1986,pp Youcef-oumi K and Shortlidge C, Control of robotic manipulators using time delay, IEEE International Conference on Robotics and Automation CA, 1991, pp Yoo, D and Chung, J, A variable structure control law with simple adaptation laws for upper bounds on the norm of uncertainties, IEEE rans Automat Cont, Vol 37, 1992, pp Youcef-oumiK and Ito, A ime Delay controller for systems with unknown dynamics, Journal of Dynamic systems, measurement and control, Vol 112, arch 1990, pp Youcef-oumiK and S Reddy, Analysis of Linear ime-invariant systems with ime Delay, rans of ASE, Vol 114, December 1992, pp Chanderasekhar was ech student of National Institute of echnology, Kurukshetra, Haryana India LDewan LISE, LSSI Assistant Professor Department of ElectEngineering National Institute of echnology Kurukshetra India l_dewanin@yahoocom Received ech and PhD degrees in Robust control from Kurukshetra University Kurukshetra in 1987 and 2001 respectively She joine the National Institue of echnology in 1984 She is presently teaching Digital Signal Processing and Optimal Control Her areas of interest are Robust control, Identification and Instrumentation and control